关于Hilbert符号的讨论

On the Discussion of Hilbert Symbol

在乘群k*中,希尔伯特定义了一种两个元素a,b之间的运算(a,b),称为希尔伯特符号,在此我们利用已经推得的(a,b)简单运算公式及性质,采用指数α,β,仅由它们的模2的剩余类决定这一特性,将α,β分成3种情形,根据乘群k*的性质、Legendre符号性质及p进方程的理论,对它的计算进行讨论.

In the multiplicative group, Hilbert definites an operation between a and b, and it denotes by the so-called Hilbert symbol （ a, b ）. In this paper , we will deduce the calculating formulas of Hilbert symbol （ a, b ） by using some results in the following： the characters of the exponents α and β are decided only by their residue class modul 2, the properties of multiplicative group and Legendre＇ s symbol, the theories of p-adic equation and so forth.